Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T06:07:50.393Z Has data issue: false hasContentIssue false
  • English
  • Français

Towards Motivic Quantum Cohomology of M̄0,S

Published online by Cambridge University Press:  19 December 2013

Yuri I. Manin  and
Maxim Smirnov
Yuri I. Manin
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany ([email protected])
Maxim Smirnov
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany ([email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We explicitly calculate some Gromov–Witten correspondences determined by maps of labelled curves of genus 0 to the moduli spaces of labelled curves of genus 0. We consider these calculations as the first step towards studying the self-referential nature of motivic quantum cohomology.

Keywords

moduli spacesGromov–Witten theoryquantum cohomologyPrimary 14N3514D22
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 1 , February 2014 , pp. 201 - 230
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Bayer, A., Semisimple quantum cohomology and blow-ups, Int. Math. Res. Not. 2004(40)(2004), 2069–2083.Google Scholar
2.Bayer, A. and Manin, Yu., (Semi)simple exercises in quantum cohomology, in The Fano Conference Proceedings (ed. Collino, A., Conte, A. and Marchisio, M.), pp. 143–173 (Universitá di Torino, 2004).Google Scholar
3.Bayer, A. and Manin, Yu., Stability conditions, wall-crossing and weighted Gromov–Witten invariants, Moscow Math J. 9(1) (2009), 3–32.CrossRefGoogle Scholar
4.Behrend, K., Gromov–Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601–617.CrossRefGoogle Scholar
5.Behrend, K., The product formula for Gromov–Witten invariants. J. Alg. Geom. 8(3) (1999), 529–541.Google Scholar
6.Behrend, K., Algebraic Gromov–Witten invariants, in New trends in algebraic geometry, London Mathematical Society Lecture Note Series, Volume 264, pp. 19–70 (Cambridge University Press, 1999).Google Scholar
7.Behrend, K. and Manin, Yu., Stacks of stable maps and Gromov–Witten invariants, Duke Math. J. 85(1) (1996), 1–60.Google Scholar
8.Bryan, J. and Karp, D., The closed topological string via the Cremona trasform, J. Alg. Geom. 14 (2005), 529–542.Google Scholar
9.Castravet, A.-M., The Cox ring of , Trans. Am. Math. Soc. 361(7) (2009), 3851–3878.CrossRefGoogle Scholar
10.Castravet, A.-M. and Tevelev, J., Hypertrees, projections, and moduli of stable rational curves, J. Reine Angew. Math. 2013(675) (2012), 121–180.Google Scholar
11.Farkas, G. and Gibney, A., The Mori cones of moduli spaces of pointed curves of small genus, Trans. Am. Math. Soc. 355(3) (2003), 1183–1199.CrossRefGoogle Scholar
12.Fulton, W. and MacPherson, R., A compactification of configuration spaces, Annals Math. 130 (1994), 183–225.CrossRefGoogle Scholar
13.Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, in Algebraic Geometry, Santa Cruz, 1995, Proceedings of Symposia in Pure Mathematics, pp. 45–96 (American Mathematical Society, Providence, RI, 1997).Google Scholar
14.Gibney, A. and MacLagan, D., Lower and upper bounds for NEF cones, Int. Math. Res. Not. 2012(14) (2011), 3224–3255.Google Scholar
15.Grothendieck, A., Éléments de géométrie algébrique: étude locale des schémas et des morphismes de schémas, Publ. Math. IHES 28 (1966).Google Scholar
16.Hacking, P., The moduli space of curves is rigid, Alg. Num. Theory 2(7) (2008), 809–818.Google Scholar
17.Hassett, B. and Tschinkel, Yu., On the effective cone of the moduli space of pointed rational curves, in Topology and Geometry: Commemorating SISTAG, Singapore International Symposium in Topology and Geometry, Contemporary Mathematics, Volume 314, pp. 83–96 (American Mathematical Society, Providence, RI, 2002).Google Scholar
18.Hu, J., Gromov–Witten invariants of blow-ups along points and curves, Math. Z. 233 (2000), 709–739.CrossRefGoogle Scholar
19.Hu, J., Quantum cohomology of blow-ups of surfaces and its functoriality property, Acta Math. Sci. 26(4) (2006), 735–743.CrossRefGoogle Scholar
20.Iritani, H., Convergence of quantum cohomology by quantum Lefschetz, J. Reine Angew. Math. 2007(610) (2007), 29–69.Google Scholar
21.Kaufmann, R., The intersection form in H* and the explicit Künneth formula in quantum cohomology, Int. Math. Res. Not. 19 (1996), 929–952.CrossRefGoogle Scholar
22.Keel, S., Intersection theory of moduli spaces of stable n-pointed curves of genus zero, Trans. Am. Math. Soc. 330 (1992), 545–574.Google Scholar
23.Keel, S. and McKernan, J., Contractible extremal rays on , eprint (arXiv:alg-geom/9607009, 1996).Google Scholar
24.Kontsevich, M. and Manin, Yu., Gromov–Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164 (1994), 525–562.CrossRefGoogle Scholar
25.Kontsevich, M. and Manin, Yu., Quantum cohomology of a product, Invent. Math. 124 (1996), 313–339.CrossRefGoogle Scholar
26.Kontsevich, M. and Manin, Yu., Relations between the correlators of the topological sigma-model coupled to gravity, Commun. Math. Phys. 196 (1998), 385–398.Google Scholar
27.Lee, Y.-P., Lin, H.-W. and Wang, C.-L., Flops, motives and invariance of quantum rings, Annals Math. 172 (2010), 243–290.CrossRefGoogle Scholar
28.Manin, Yu., Frobenius manifolds, quantum cohomology, and moduli spaces, Colloquium Publications, Volume 47 (American Mathematical Society, Providence, RI, 1999).Google Scholar
29.Manolache, Ch., Virtual pull-backs, J. Alg. Geom. 21 (2012), 201–245.Google Scholar
30.Manolache, Ch., Virtual push-forwards, Geom. Topol. 16 (2012), 2003–2036.CrossRefGoogle Scholar
31.Maulik, D. and Pandharipande, R., A topological view of Gromov–Witten theory, Topology 45 (2006), 887–918.Google Scholar